Distribution of velocities

Maxwell s Distribution of Velocities.—lieturning to our distribution law (1.5), let us first consider the case where there is no potential energy, so that the distribution is independent of position in space. Then the fraction of all molecules for which the momenta lie in dpx dpy dpt is 

Starting with je au2 du, for the even powers of n, or with fue au du for the odd powers, each integral can be found from the one above it by differentiation with respect to —a, by means of which the table can be extended. To get the integral from — co to oo, the result with the even powers is twice the integral from 0 to oo f and for the odd powers of course it is zero. 

Using the integrals (2.3), the quantity (2.2) becomes (2wmkT) K Thus we may rewrite Eq. (2.1) as follows the fraction of molecules with momentum in the range dpx dpu dpz is 

Equation (2.4) is one form of the famous Maxwell distribution of velocities. 

Often it is useful to know, not the fraction of molecules whose vector velocity is within certain limits, but the fraction for which the magnitude of the velocity is within certain limits. Thus let v be the magnitude of the velocity  

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It is important to recognize the approximations made here the electric field is supposed to be sulficiently small so that the equilibrium distribution of velocities of the ions is essentially undisturbed. We are also assuming that the we can use the relaxation approximation, and that the relaxation time r is independent of the ionic concentration and velocity. We shall see below that these approximations break down at higher ionic concentrations a primary reason for this is that ion-ion interactions begin to affect both x and F, as we shall see in more detail below. However, in very dilute solutions, the ion scattering will be dominated by solvent molecules, and in this limiting region A2.4.31 will be an adequate description. 

This rate constant refers to reactants which all move with a velocity v whereas the usual situation is such that we have a Boltzmaim distribution of velocities. If so then the rate constant is just the average of (A3.11.173) over a Boltzmaim distribution Pg ... 

The initial velocities may also be chosen from a uniform distribution or from a simp Gaussian distribution. In either case the Maxwell-Boltzmann distribution of velocities usually rapidly achieved. 

If Restart is not checked then the velocities are randomly assigned in a way that leads to a Maxwell-Boltzmann distribution of velocities. That is, a random number generator assigns velocities according to a Gaussian probability distribution. The velocities are then scaled so that the total kinetic energy is exactly 12 kT where T is the specified starting temperature. After a short period of simulation the velocities evolve into a Maxwell-Boltzmann distribution. 

Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components i , and v.. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. 

Maxwell found that he could represent the distribution of velocities statistically by a function, known as the Maxwellian distribution. The collisions of the molecules with their container gives rise to the pressure of the gas. By considering the average force exerted by the molecular collisions on the wall, Boltzmann was able to show that the average kinetic energy of the molecules was... 

The distribution of velocity components (radial, tangential and axial) under conditions of mixing with baffles in comparison with the conditions of vortex formation is presented in Figure 12. The dashed lines in Figure 12 indicate non-baffled conditions. Comparison of the non-baffled and fully baffled velocity curves (solid line) leads to the following set of conclusions on vortex suppression when dealing with perfectly miscible liquids ... 

As outlined earlier, in multizone models, perfect mixing is assumed in the individual zone. The spatial distribution of velocities, contaminant concentrations, and air temperatures in a zone can be determined only by using CFD. On the other hand, wind effects are easily accounted for in multizone models, and unsteady-state simulation is normally performed. On the combined use of the two methods, see Schaelin et al.--... 

Tor pure plumes the distribution of velocity, buoyancy, and concentration is the same. 

For the distribution of velocities in the gas, in any given plane we have, according to Maxwell s distribution law ... 

Most theoretical studies of heat or mass transfer in dispersions have been limited to studies of a single spherical bubble moving steadily under the influence of gravity in a clean system. It is clear, however, that swarms of suspended bubbles, usually entrained by turbulent eddies, have local relative velocities with respect to the continuous phase different from that derived for the case of a steady rise of a single bubble. This is mainly due to the fact that in an ensemble of bubbles the distributions of velocities, temperatures, and concentrations in the vicinity of one bubble are influenced by its neighbors. It is therefore logical to assume that in the case of dispersions the relative velocities and transfer rates depend on quantities characterizing an ensemble of bubbles. For the case of uniformly distributed bubbles, the dispersed-phase volume fraction O, particle-size distribution, and residence-time distribution are such quantities. 

If a fluid passes at right angles across a single tube, the distribution of velocity around the tube will not be uniform. In the same way the rate of heat flow around a hot pipe across which air is passed is not uniform but is a maximum at the front and rear, and a minimum at the sides, where the rate is only some 40 per cent of the maximum. The general picture is shown in Figure 9.26 but for design purposes reference is made to the average value. 

When a fluid is flowing under streamline conditions over a surface, a forward component of velocity is superimposed on the random distribution of velocities of the molecules, and movement at right angles to the surface occurs solely as a result of the random motion... 

Using the system (9.15-9.17) we determine the distribution of velocity, temperature and pressure within the liquid and vapor domains. We render the equations dimensionless by the following characteristic scales l,o for velocity, 7l,o for temperature, Pl,o for density, Pl,q for pressure, Pl,o Lo f " force and L for length... 

Often, we will be interested in how the velocities of molecules are distributed. Therefore we need to transform the Boltzmann distribution of energies into the Maxwell-Boltzmann distribution of velocities, thereby changing the variable from energy to velocity or, rather, momentum (not to be confused with pressure). If the energy levels are very close (as they are in the classic limit) we can replace the sum by an integral ... 

However, it was Maxwell in 1848 who showed that molecules have a distribution of velocities and that they do not travel in a direct line. One experimental method used to show this was that ammonia molecules are not detected in the time expected, as derived from their calculated velocity, but arrive much later. This arises l om the fact that the ammonia molecules tnterdiffuse among the air moixules by intermolecular collisions. The molecular velocity calculated for N-ls molecules from the work done by Joule in 1843 was 5.0 xl02 meters/sec. at room temperature. This implied that the odor of ammonia ought to be detected in 4 millisec at a distance of 2.0 meters from the source. Since Maxwell observed that it took several minutes, it was fuUy obvious that the molecules did not travel in a direct path. 

We use a conventional experimental design, in that fluid is flowed at a constant flow rate through a sample. We measure the pressure drop and the distribution of velocity within the sample, as described in the following section. We then estimate the permeability distribution from the measured data, as described in Section 4.1.4.2.2. 

Fig. 4.6.7 Projections along the secondary diagonal from the 2D VEXSY experiments presented partly in Figure 4.6.5 and 4.6.6. (a) Distribution of velocity change obtained among others from Figure 4.6.5 (a, d) of the SMC module, (b) Distribution of velocity change obtained among others from Figure 4.6.6(a, d) of the SPAN module, (c) Three out of six distributions presented in (a) and (b) are displayed as the distribution of acceleration, which is obtained by dividing the velocity...

Fig. 5.1.8 MRM maps of velocity (bottom row) resolution is 54.7 im per pixel (128 x 128 -and T2 magnetic relaxation (top row) as a pixels) in plane, so the data reflect pore scale function of biofilm growth time (left to right), spatial distributions of velocity over a 1000-pm Day 1 shows the clean porous media. Spatial slice and biomass over a 200-gm slice.

These methods hardly take spatial distributions of velocity field and chemical species or transient phenomena into account, although most chemical reactors are operated in the turbulent regime and/or a multiphase flow mode. As a result, yield and selectivity of commercial chemical reactors often deviate from the values at their laboratory or pilot-scale prototypes. Scale-up of many chemical reactors, in particular the multiphase types, is still surrounded by a fame of mystery indeed. Another problem relates to the occurrence of thermal runaways due to hot spots as a result of poor local mixing effects. 

The frequency with which the transition state is transformed into products, iT, can be thought of as a typical unimolecular rate constant no barrier is associated with this step. Various points of view have been used to calculate this frequency, and all rely on the assumption that the internal motions of the transition state are governed by thermally equilibrated motions. Thus, the motion along the reaction coordinate is treated as thermal translational motion between the product fragments (or as a vibrational motion along an unstable potential). Statistical theories (such as those used to derive the Maxwell-Boltzmann distribution of velocities) lead to the expression ... 

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